    
Thus due to (a) it remains to show that \(L\subset \overline{\langle M\rangle }\) . So for \(U\)  open bounded consider the set \(\mathcal {B}:=\lbrace B: B\subset U, B\textnormal { closed ball}\rbrace \) . Then according to Vitali's covering theorem [1]} there are countably many mutually disjoint \(B_i\in \mathcal {B}\)  with \(\mu (U\setminus \bigcup _iB_i)=0\) . Hence, for \(\varepsilon >0\)  there is \(n\in \mathbb {N}\)  with \(\mu (U\setminus V) <\varepsilon ^p\)  for \(V:=\bigcup _{i=1}^nB_i\) . Consequently, \(\parallel \!\!1_U-\sum _{i=1}^n1_{B_i}\!\!\parallel _p=\parallel \!\!1_U-1_V\!\!\parallel _p=\parallel \!\!1_{U\setminus V}\!\!\parallel _p=\mu (U\setminus V)^{1/p}<\varepsilon \) , accomplishing the proof.\(\Box \)
